I've had this experience before twice, both times required a different solution, Your case might be different than mine was, but I'll provide my two methods of resolving the issue.
Originally Posted by bellizima
A) Go into your Email accounts and double check your password. Did you recently change your account password and forget to update it on your device? Try logging into Gmail web access with the password and see if it works.
B) For some reason, I had to go through the CAPTCHA unlock process to get my iPhone to access my Gmail account again, which you can do on this page, Put in one of your Gmail accounts, your password, and the text in the CAPTCHA and click "Unlock" and then try accessing your account(s) on your iPhone again.